LFD Book Forum Problem 2.13
 User Name Remember Me? Password
 Register FAQ Calendar Mark Forums Read

 Thread Tools Display Modes
#1
07-10-2017, 09:58 AM
 RicLouRiv Junior Member Join Date: Jun 2017 Posts: 7
Problem 2.13

I'm having some trouble with Problem 2.13 -- especially Part C. Let me walk through the other parts first...

For Part A, if the VC dimension of a hypothesis set is , then you need at least hypotheses to run the points through in order to get them to shatter. So, .

For Part B, at worst, your hypothesis set contains no, or perhaps 1, hypothesis. In this event, you have . I claim you can't have a VC dimension of more than . Assume to the contrary that you could shatter points in the intersection; hence, you can find points that are shattered by . But since these hypotheses belong to every individual , it means that you can shatter these points in the that has the minimum dimension -- a contradiction. So, .

For Part C, it seems obvious that your lower bound is always the maximum VC dimension from the individual sets: -- simply because the hypotheses that shatter points in the corresponding set will carry through to the union.

The upper bound is harder. I can think of simple cases (i.e., 2 hypothesis spaces, each with 2 hypotheses in it) where . While looking ahead a little at Problem 2.14, I'm trying to see if I can find a case where . I can't find one, so I'm just hoping that is a break point. I want to argue as follows: present me with points that are shattered, and it must be the case that either or shattered or points, which is a contradiction. So, given the points, if you restrict to the subset of the union that contains all hypotheses in , by assumption it can't shatter more than points, so some dichotomy is missing...but then I get stuck.

Any hints?
#2
07-12-2017, 02:29 AM
 RicLouRiv Junior Member Join Date: Jun 2017 Posts: 7
Re: Problem 2.13

The internet helped me out here. Using growth functions (duh), you can show that , and so is a break point for the union.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home General     General Discussion of Machine Learning     Free Additional Material         Dynamic e-Chapters         Dynamic e-Appendices Course Discussions     Online LFD course         General comments on the course         Homework 1         Homework 2         Homework 3         Homework 4         Homework 5         Homework 6         Homework 7         Homework 8         The Final         Create New Homework Problems Book Feedback - Learning From Data     General comments on the book     Chapter 1 - The Learning Problem     Chapter 2 - Training versus Testing     Chapter 3 - The Linear Model     Chapter 4 - Overfitting     Chapter 5 - Three Learning Principles     e-Chapter 6 - Similarity Based Methods     e-Chapter 7 - Neural Networks     e-Chapter 8 - Support Vector Machines     e-Chapter 9 - Learning Aides     Appendix and Notation     e-Appendices

All times are GMT -7. The time now is 09:35 AM.

 Contact Us - LFD Book - Top